Problem 84
Question
find the sum of the first ten terms of the sequence $$a+b, a^{2}+2 b, a^{3}+3 b, a^{4}+4 b, \ldots$$
Step-by-Step Solution
Verified Answer
\( a + a^2 + a^3 + a^4 + a^5 + a^6 + a^7 + a^8 + a^9 + a^{10} + 55b \)
1Step 1: Identify the Sequence Pattern
The sequence follows the pattern: the nth term is expressed as \( a^n + n \, b \). Identifying this will help us find the first ten terms of the sequence.
2Step 2: Write Down the First Ten Terms
Substitute \( n \) with the numbers from 1 to 10 to find the first ten terms: 1. \( a^1 + 1b = a + b \)2. \( a^2 + 2b \)3. \( a^3 + 3b \)4. \( a^4 + 4b \)5. \( a^5 + 5b \)6. \( a^6 + 6b \)7. \( a^7 + 7b \)8. \( a^8 + 8b \)9. \( a^9 + 9b \)10. \( a^{10} + 10b \).
3Step 3: Sum the Terms
Add the terms together to find the sum of the first ten terms:\[ (a + b) + (a^2 + 2b) + (a^3 + 3b) + (a^4 + 4b) + \ldots + (a^{10} + 10b)\]
4Step 4: Simplify the Expression
Combine like terms in the sum:- Combine all the \( a^n \) terms: \[ a + a^2 + a^3 + a^4 + a^5 + a^6 + a^7 + a^8 + a^9 + a^{10} \]- Combine all the \( n \, b \) terms (which form an arithmetic series): \[ b(1 + 2 + 3 + \ldots + 10) \].
5Step 5: Evaluate the Arithmetic Series
Calculate the sum of the arithmetic series \( 1 + 2 + 3 + \ldots + 10 \). The formula for the sum of the first \( n \) natural numbers is: \[ \frac{n(n+1)}{2} \]Substitute \( n = 10 \):\[ \frac{10 \times 11}{2} = 55 \].
6Step 6: Final Calculation
Add the geometric sequence sum to the arithmetic series sum:\[ a + a^2 + a^3 + a^4 + a^5 + a^6 + a^7 + a^8 + a^9 + a^{10} + 55b \].
7Step 7: Write Out the Final Answer
The sum of the first ten terms of the sequence is:\[ a + a^2 + a^3 + a^4 + a^5 + a^6 + a^7 + a^8 + a^9 + a^{10} + 55b \].
Key Concepts
Arithmetic SeriesGeometric SequenceSum of Series
Arithmetic Series
An arithmetic series is formed by the sum of the terms of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference and is usually denoted by the letter \( d \).
For example, if you have a sequence like \( 2, 4, 6, 8, 10 \), each number is obtained by adding the common difference of 2 to the term before it. The series derived from this sequence would be \( 2 + 4 + 6 + 8 + 10 \).
To find the sum of an arithmetic series, we use the formula:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
Where:
In our original problem, the terms \( b(1 + 2 + 3 + \ldots + 10) \) form an arithmetic series with a sum calculated as \( 55b \). This was derived using the formula for the sum of the first \( n \) natural numbers.
For example, if you have a sequence like \( 2, 4, 6, 8, 10 \), each number is obtained by adding the common difference of 2 to the term before it. The series derived from this sequence would be \( 2 + 4 + 6 + 8 + 10 \).
To find the sum of an arithmetic series, we use the formula:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
Where:
- \( S_n \) is the sum of the series.
- \( n \) is the number of terms.
- \( a_1 \) is the first term.
- \( a_n \) is the nth term.
In our original problem, the terms \( b(1 + 2 + 3 + \ldots + 10) \) form an arithmetic series with a sum calculated as \( 55b \). This was derived using the formula for the sum of the first \( n \) natural numbers.
Geometric Sequence
A geometric sequence is a sequence in which each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This concept is crucial in understanding patterns where growth or decay multiplies rather than adds.
For example, in the sequence \( 3, 6, 12, 24, 48 \), each term after the first is the previous term multiplied by the common ratio 2.
In mathematical terms, a geometric sequence is typically represented as:
For the original problem we're analyzing, notice the terms such as \( a, a^2, a^3, \ldots, a^{10} \). Here, the exponent on \( a \) increases progressively by one with each term, resembling a geometric pattern. This pattern is reminiscent of a geometric sequence where each term is the previous term multiplied by a constant (in this case, the base number \( a \)). This understanding is crucial because the geometric progression \( a, a^2, a^3, \ldots \) forms part of the term that we need to sum up in the original exercise.
For example, in the sequence \( 3, 6, 12, 24, 48 \), each term after the first is the previous term multiplied by the common ratio 2.
In mathematical terms, a geometric sequence is typically represented as:
- \( a, ar, ar^2, ar^3, \ldots \)
For the original problem we're analyzing, notice the terms such as \( a, a^2, a^3, \ldots, a^{10} \). Here, the exponent on \( a \) increases progressively by one with each term, resembling a geometric pattern. This pattern is reminiscent of a geometric sequence where each term is the previous term multiplied by a constant (in this case, the base number \( a \)). This understanding is crucial because the geometric progression \( a, a^2, a^3, \ldots \) forms part of the term that we need to sum up in the original exercise.
Sum of Series
In mathematics, the sum of a series refers to adding the terms of a sequence together to get a single expression or number. Understanding how to find this sum is essential when dealing with sequences in mathematics, particularly arithmetic and geometric sequences.
An arithmetic series, as described earlier, uses the formula for the sum of an initial series of terms. The efficiency of these formulas simplifies the process significantly, especially for larger numbers of terms.
Similarly, the sum of a geometric series can also be calculated using a formula. For a finite geometric series, the sum is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Where:
In the original exercise, calculating the sum of the sequence involves recognizing and separating the arithmetic series part \( b(1 + 2 + 3 + \ldots + 10) \) and the geometric-like sequence \( a, a^2, a^3, \ldots \), then adding these series together to find the overall sum of all the terms.
An arithmetic series, as described earlier, uses the formula for the sum of an initial series of terms. The efficiency of these formulas simplifies the process significantly, especially for larger numbers of terms.
Similarly, the sum of a geometric series can also be calculated using a formula. For a finite geometric series, the sum is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Where:
- \( S_n \) is the sum of the series.
- \( a \) is the first term.
- \( r \) is the common ratio.
- \( n \) is the number of terms.
In the original exercise, calculating the sum of the sequence involves recognizing and separating the arithmetic series part \( b(1 + 2 + 3 + \ldots + 10) \) and the geometric-like sequence \( a, a^2, a^3, \ldots \), then adding these series together to find the overall sum of all the terms.
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Problem 83
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